Feynman-Kac formulae

January 27, 2021 — January 27, 2021

Bayes

Monte Carlo

probabilistic algorithms

probability

state space models

statistics

swarm

time series

There is a mathematically rich theory about particle filters, and the central tool to make that go seems to be Feynman-Kac formulae. The notoriously abstruse Del Moral (2004);Doucet, Freitas, and Gordon (2001) are regarded as the unifying introduction to these formulae, whatever they are. A diligent student will supposedly make consistent the diffusion processes and Feynman-Kac formulae and “propagation of chaos” and their delicate relationships. I will get around to understanding them myself eventually, maybe?

Related, apparently: Backward SDEs.

1 References

Beck, E, and Jentzen. 2019. “Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-Order Backward Stochastic Differential Equations.” Journal of Nonlinear Science.

Cérou, Moral, Furon, et al. 2011. “Sequential Monte Carlo for Rare Event Estimation.” Statistics and Computing.

Chan-Wai-Nam, Mikael, and Warin. 2019. “Machine Learning for Semi Linear PDEs.” Journal of Scientific Computing.

Chopin, and Papaspiliopoulos. 2020. An Introduction to Sequential Monte Carlo. Springer Series in Statistics.

Chuang. 2010. “The Feynman-Kac Formula:Relationships Between Stochastic DifferentialEquations and Partial Differential Equations.”

Del Moral. 2004. Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications.

Del Moral, and Doucet. 2010. “Interacting Markov Chain Monte Carlo Methods for Solving Nonlinear Measure-Valued Equations.” The Annals of Applied Probability.

Del Moral, Hu, and Wu. 2011. On the Concentration Properties of Interacting Particle Processes.

Del Moral, and Miclo. 2000. “Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering.” In Séminaire de Probabilités XXXIV. Lecture Notes in Mathematics 1729.

Doucet, Freitas, and Gordon. 2001. Sequential Monte Carlo Methods in Practice.

Doucet, Godsill, and Andrieu. 2000. “On Sequential Monte Carlo Sampling Methods for Bayesian Filtering.” Statistics and Computing.

E, Han, and Jentzen. 2017. “Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations.” Communications in Mathematics and Statistics.

Han, Jentzen, and E. 2018. “Solving High-Dimensional Partial Differential Equations Using Deep Learning.” Proceedings of the National Academy of Sciences.

Hartmann, Richter, Schütte, et al. 2017. “Variational Characterization of Free Energy: Theory and Algorithms.” Entropy.

Hutzenthaler, Jentzen, Kruse, et al. 2020. “Overcoming the Curse of Dimensionality in the Numerical Approximation of Semilinear Parabolic Partial Differential Equations.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

Hutzenthaler, and Kruse. 2020. “Multilevel Picard Approximations of High-Dimensional Semilinear Parabolic Differential Equations with Gradient-Dependent Nonlinearities.” SIAM Journal on Numerical Analysis.

Kebiri, Neureither, and Hartmann. 2019. “Adaptive Importance Sampling with Forward-Backward Stochastic Differential Equations.” In Stochastic Dynamics Out of Equilibrium. Springer Proceedings in Mathematics & Statistics.

Nualart, and Schoutens. 2001. “Backward Stochastic Differential Equations and Feynman-Kac Formula for Lévy Processes, with Applications in Finance.” Bernoulli.

Nüsken, and Richter. 2021. “Solving High-Dimensional Hamilton–Jacobi–Bellman PDEs Using Neural Networks: Perspectives from the Theory of Controlled Diffusions and Measures on Path Space.” Partial Differential Equations and Applications.

Papanicolaou. 2019. “Introduction to Stochastic Differential Equations (SDEs) for Finance.” arXiv:1504.05309 [Math, q-Fin].